We proved that $G_\pi = (V, E_\pi)$, where $E_\pi = \{ (\pi(v),v) : v\in V_\pi\setminus\{s\} \}$ is a tree if and only if no negative cycles are reachable from $s$. Otherwise, there's a cycle in $G_\pi\subseteq G$, and its weight must be negative.